RANDOM ON

I nternat. J. Math. Math. Si.

Vol. 4 No. 4 (1981)

745-752

745

ON THE ALMOST SURE CONVERGENCE OF WEIGHTED SUMS OF RANDOM ELEMENTS IN D[0,1]

R.L. TAYLOR and C.A. CALHOUN

Department of Mathematics and Statistics University of South Carolina

Columbia, S. C. 29208

(Received June 26, 1980 and in revised form February 17, 1981)

ABSTRACT. Let

{w }

ⁿ be a sequence of positive constants and Wⁿ

Wl+...+

^{wn where}

/ and /

n w

n_W

_n

.

^Let

{Wn ^}

^{be a sequence of} independent random elements in W

D[O,1]. The almost sure convergence of W-1

n Wk

^is established under certain n k=l

integral conditions and growth conditions on the weights

{w }.

The results are n

shown to be substantially stronger than the weighted sums convergence results of Taylor and Daffer (1980) and the strong laws of large numbers of Ranga Rao (1963) and Daffer and Taylor (1979).

KEY WORDS AND PHRASES. Weighted Sums, Random Elements

in

D[O, I], Strong Laws of Lge Numbr -ntegral Conditions, ^and Almost Sure Convergence.

1980

THEMATICS SUBJECT CLASSIFICATION CODES. Primay 60B12; Secondary 60F15.

i INTRODUCTION

The convergence of weighted sums of random elements in

D[0,1]

was obtained by Taylor and Daffer (1980) using a number of conditions such as convex tightness, moment conditions, and others. In this paper both the moment conditions and the tightness conditions are substantially relaxed in obtaining the almost sure con- vergence of weighted sums of random elements in

D[0,1].

In addition, a brief discussion of the necessary integral conditions will be included which will de- lineate these results and previous work.

Let

D[0,1](--D)

denote the space of real-valued functions on

[0,i]

which are right continuous and possess left-hand limits for each t [0,i]. Let the linear space D be equipped with the topology generated by the Skorohod metric d and let

II II

^{denote the uniform norm,}

llxll

^sup

^Ix(t)

^{for x D. Next,}

t[0,1]

let

(, A,

P) denote a probability space. A random element X in D is a function X: ^/ D which is measurable with respect to the Borel sets of the Skorohod topol- ogy. Random elements in D are characterized by the property that X(t) is a random variable for each tE

[0,I].

^The expected value EX D can be defined pointwise by (EX)(t) m(x(t)) ^for each tE [0,I] when E

llXll

^<

.

Detailed geometric and probabilistic properties of the space D with the uniform norm and the Skorohod metric can be found in Billingsley

(1968),

pp. 109-153 and Taylor (1978), pp. 153-184.

2. INTEGRAL CONDITIONS.

In obtaining the strong law of large numbers for independent, identically distributed random elements in D, Ranga Rao (1963) used the following crucial

LEMMA i. If X is a random element in D with E

llxlloo ^{< oo,}

^{then for each E > 0}

there exists a partition 0 t O < t

I

<...< tm

^{i of [0,i] such that}

max sup

ElX(t)

X(s)

-< e.

_(2.1)

l_<i<m

ti_l_<t, s<ti

In extending inequality (2.1) to a sequence of random elements and in trying to obtain strong laws of large numbers for non-identically distributed random elements, the following formulation of integral properties were motivated. In each of the following properties and in Lemma i, the partition

{tl,...

^tm ^depends on E.

DEFINITION. A sequence of random elements

{X

_{in D is said} to satisfy (a) n

property (RT) if for each e

>

0 there exists a partition 0 t

0<

^tI

<...< tm

ⁱ

of

[0,13

such that

CONVERGENCE OF WEIGHTED SUMS OF RANDOM ELEMENTS 747 max

E[IX___(ti+l

^{O) X}

^(t)I]

^e

sup n i

n 0-<i<m-i

(2.2)

(b) property (mT) if for each e > 0 there exists a partition 0 t

o ^<

^tI

<...<

t i of [0,i] such that m

sup max E[ sup

IX n(t)

^X

n(t i) ^I]

^<

^e.

n ^0_<im-I

ti_<t<ti+

_I

(2.3)

(c) property (T) if for each e > 0 there exists a (Skorohod) compact set K such that

sup E

fIX

_n

_I[X K]II

^{< e. (2.4)}

n n

Clearly, property (mT) implies property (RT). Also, identically distributed random elements

{Xn ^}

^{with E}

IIXII<

^{satisfy (RT)} obviously and satisfy (mT) by Lemma 2 since (T) follows from the tightness of identical distributions.

LEMMA 2. If

{X }

satisfies property (RT) and property (T), then

{X

satis-

n n

fies property (mT).

PROOF. Let e > 0 be gdven, Choose K (Skorohod) compact such that

sup E

fiX

_n

I[X K]II

^{< e/6. (2.5)"}

n n

Since K is compact, there exists > 0 such that

Ix(t)

^x(s)

^<- Ix(u-0)

^x(s)

+

e/3 (2.6)

for each x E K whenever 0 s s S t < u

<

s

+

6.

By (RT) choose a partition

{t l,...,tm ^}

^of

^[0,I]

^{such that}

max

E[[X"(ti+ILL

⁰⁾

L-X"(ti) ^l]

^{< g/3. (2.7)}

sup

n i

Without loss of generality ^it can be assumed that

ti+.

^{tol < for all}

i 0,i m-l. Thus, from (2.6) and (2.5) sup max E[ sup

IX n(t)

^X

n(t i) ^l]

n i t

i<t.<ti+l

by (2.7).

< sup maxn i E[

ti-

<t<_supti+l

IXn(t) Xn(ti)]l[x

n K]

+

sup max E[ sup

01Xn(t)

^X_n

^(t)I

^I

n i

ti<t<ti+

I

IX

^K]

n

< sup_n max_i

E[(IX_(ti+

I, 0)

Xn ^{(ti) +}

^{e/3) I}

Ix

_n ^K]

+

sup_n

E[211

^X

nll ^l[x

_n ^K]]

-<

sup max

E[IXn(ti+l-0) Xn(ti) II

n i

Ix

^K]

n

]

+ el3 + el3

III

Property (T) has been used in several laws of large numbers for random ele- ments in Banach spaces to eliminate the assumption of identical distributions.

In obtaining laws of large numbers for D, Daffer and Taylor (1980) required con- vex tightness or that the (Skorohod) compact set in (2.4) also be convex. In particular, the previous convex tightness condition can be partially formulated by the integral condition

sup E [max sup

IXn(t)

^X

n(t i) ^{l] <- e.}

n i

ti-<t<ti+

I

(2.8)

Property (mT) is clearly much more general than

(2.8),

and convergence results in the next section using property (mT) are a marked improvement over the results of Taylor and Daffer (1980) and include the results for identical distributions.

3. ALMOST SURE CONVERGENCE.

Let

{

_w

k}

^{denote a sequence of positive constants. Define}

W w

I

+...+

w S

n

In

ⁱ

n n k=I

WkXk,

^{and N(x) card}

^{n: W-n Wn ^x}.

^Random

variables

{X }

are said to have uniformly bounded ^tail probabilities if there n

exists a nonnegative real-valued random variable X such that

P[IXnl

^{> t]}

^-<

^P[X

^>

^t] for each n and for all t R. This property is denoted by

{X

c X. The following real-valued result can be obtained from the geometric

n

CONVERGENCE OF WEIGHTED SUMS OF RANDOM ELEMENTS 749 Banach space results of Howell, Taylor and Woyczynski (to appear) and will be used in establishing the almost sure convergence of weighted sums of random elements in D.

THEOREM i. Let

{X }

be independent (real-valued) random variables such that n

{X }

e X, EN(X) <

=,

and for some i < p < 2 n

f0

^{tP-I P[X}

^>

^t]

ft ^y-(P+l)

^{N(y)dy dt <}

^=.

If W_n ^/

,

^then

W-Is

_{n n} ^-c_n ^{/0. a.s.}

where

cⁿ

W-i

ⁿ

^’k=l

ⁿ

^{We m( l[iXk}

^_<

w ^{I Wk])}

REMARK. If

Wn

-i

Wn

^/

’

^{EX <}

^,

^{and EX}

^{I EXn}

^{for all n, then}

Wn

-i

Sn EXI

^{/ 0}

since

EX _{EX I}

w-I

n n

[IXnl

^< _n ^{W ]}_n

;

w-lW

n n

tdP[IXnl ^-<

^t]dt

n n

tdP[IXnl

^{> t] <}

w-

_{n n} ^P[X

^> w-

_{n n}^]

⁺ _f=o

^{P[X > t]dt}

w-iW

n n

/ 0 as n

-+=#.

THEOREM 2. Let

{X }

be independent random elements in D such that n

(i) property (mT) is satisfied, (ii)

{ IIXnll ^}

^{X and EN(X)}

^< ,

(iii)

EXn

^{EX1 for all n, and (iv) for some i _< p _< 2 Condition (3.1) holds.}

If W_n ^/ and

w-

_{n n} ^/

^=,

^then

IIW

^{-In Sn EX}

Ill

^{/ 0 a,s,}

PROOF. By property (mT) there exists a partition 0 t O < t

I

<...< tm

ⁱ

such that

max E[ sup

[Xk(t) Xk(ti) ^{[3 e}

for each k.

O<i<m-i

ti<t<ti+ ^I

(3.2)

For each n

< ^Oi<m-imax

W-I

ⁿ

.

ⁿ

^k=lWk ti<t<ti+

^sup

^{I l(t) (tl)}

+

_Oim^max

IW ^{I [k=l}

n

^{Wk(tl) EXl(ti)}

+

max sup

IEX l(t)

^EX

l(t i) l.

^(3.3)

OSim-i

ti<t<ti+ I

Similar to the proofs in Taylor and Daffer

(1980),

the last term in (3.3) is less than

e

by property (mT). Also, noting that for each i

{(ti)}

^{X, the second}

term in (3.3) converges to 0 a.s. by Theorem i. For the first term of (3.3), define

ti<t<-ti+

¹

_(i)}

are independent random variables with zero means and For each i

{z

k

{zk(i) ^}

^{a 2X. Thus}

W-I _{in WkZk}

^{(i) / 0}

n k=l a.s.

by Theorem i, for each i 0,1,...,m-l. Also, by Theorem i

n

k(i)

lira_nsup

W llk=In ^Wk

^{Y i) < lira}_n^sup

Wnl Ik=l ^Wk

^EY

^e

^(3.5)

a.s. for each i by property (mT). Hence, lim sup

IIW

n

-lln

k=l

Wk- EXII ^-<

n

a.s.,

^(3.6)

and the proof is completed by taking a sequence of

’s

going to zero and by ex-

cluding a countable union of null sets. ///

CONVERGENCE OF WEIGHTED SUMS OF RANDOM ELEMENTS 751 To obtain the traditional strong law of large numbers, let w

k i for all k.

Then, N(y) [y] (the greatest integer function), and condition (3.1) becomes approximately

f0 ^tp-I

^P[X

^>

^t]

ft ^y-(p+l)

^{Y dy dt}

f0

^P[X

^>

^t]dt

for each p

>

i. Thus, the following corollary is immediate.

COROLLARY i. If {X

}

is a sequence of

independent

random elements in D such n

that (i) property (mT) is satisfied (li)

{X }

X and EX

< =o,

and (ill) EX EX

n n

I

then

lln

^-I

I

ⁿ_k=

_I

^Xk EX I

II

^{/ 0}

It is easy to show that

{X

EX

}

has property (mT) if

{X }

has property

n n n

(mr). Also, if for some p >

I,

^g

fIX nll ^{e -<}

^{M for all n, then}

(p+l) P[

llXnl

^{> t] <}

^t-e

^M

f ^s-

^{pM ds}

will yield a random variable X such that

{X }

_X and n

(p+l)

s-p

ds

< .

EX

fM_p ^s-

^{spM ds-- pM}

_/M_p

COROLLARY 2. If

{X n}

^{is a} sequence of independent random elements in D such that (i) property (mr) holds and (ii) for some p > i, sup E_n

llXnll

^p

^{< ==,}

^then

Corollary 2 represents a major improvement over Theorem i of Daffer and Taylor (1979) in that not only is convergence in the Skorohod topology replaced by convergence ⁱⁿ the uniform norm but the restrictive condition of

convex

tight- ness is replaced by property (mT). Finally, since identically distributed random elements with a first absolute moment satisfy property

(mT),

the following more general form of the strong law of large numbers is available.

THEOREM 3. If

{X }

are identically distributed random elements in D such n

that (i) EXI ^{exists and} EN(

llXll

^{< and (ii) condition (3.1) holds for}

llXll

^{X, then}

I[W llk_-I

n

_{Wk EXI} ^[[

^{/ 0 aoS.}

ACKNOWLEDGEMENTS. The authors are grateful ^to Peter Z. Daffer for his dis- cussions and help in the development of the integral conditions and their rela- tionships, and to the referee for helpful comments on organization of the paper.

The research was supported in part by the Air Force Office of Scientific Research under Contract number F49620-79-C-0140 and by the Research and Productive Scholar- ship Fund of the University of South Carolina.

REFERENCES

.

Billingsley, P. Convergence of Probability Measures, ^{Wiley, New} York, (1968).

2. Daffer, P. Z. and Taylor, R. L. Laws of large numbers for

D[0,1],

Annals of Probabili.ty 7 (1979), 85-95.

3. Howell, J., Taylor, R. L., and Woyczynski, W. A. Stability of linear forms in independent random variables in Banach spaces, (to appear) Proc.

Prob. in

Banac..h ^Spaces.

^III.

4. Rao, R. R. The law of large numbers for

D[0,1]-valued

^random variables,

Theory

^{of Prob. and Appl.}

_8

(1963), 70-74.

5. Taylor, R. L. Stochastic

Convergence

^of

Weighted

Sums of Random Elements in Linear

Spaces.

Lecture Notes in Mathematics, 672. Springer-Verlag, Berlin, 1978.

6. Taylor, R. L. and Daffer, P. Z. Convergence of weighted sums of random elements in

D[0,1],

J. of Mult. Anal. i0

(1980),

95-106.